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A Geometry of Music


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Table of Contents

PREFACE PART I. Theory CHAPTER 1. Five Components of Tonality 1.1 The five features. 1.2. Perception and the five features. 1.3 Four Claims. A. Harmony and counterpoint constrain each other. B. Scale, macroharmony, and centricity are independent. C. Modulation involves voice leading. D. Music can be understood geometrically. 1.4 Music, magic, and language. 1.5 Outline of the book, and a suggestion for impatient readers. CHAPTER 2. Harmony and Voice Leading 2.1 Linear pitch space. 2.2 Circular pitch-class space. 2.3 Transposition and inversion as distance-preserving functions. 2.4 Musical objects. 2.5 Voice leadings and chord progressions. 2.6 Comparing voice leadings. 2.7 Voice-leading size. 2.8 Near identity. 2.9 Harmony and counterpoint revisited. 2.10 Acoustic consonance and near-evenness CHAPTER 3. The Geometry of Chords 3.1 Ordered pitch space. 3.2 The Parable of the Ant. 3.3 Two-note chord space. 3.4 Chord progressions and voice leadings in two-note chord space. 3.5 Geometry in analysis. 3.6 Harmonic consistency and efficient voice leading. 3.7 Pure parallel and pure contrary motion. 3.8 Three-dimensional chord space. 3.9 Higher-dimensional chord spaces. 3.10 Voice leading lattices. 3.11 Triads are from Mars, seventh chords are from Venus. 3.12 Two musical geometries. 3.13 Study guide. CHAPTER 4. Scales 4.1 A scale is a ruler. 4.2 Scale degrees, scalar transposition, scalar inversion. 4.3 Evenness and scalar transposition. 4.4 Constructing common scales. 4.5 Modulation and voice leading. 4.6 Voice leading between common scales . 4.7 Two examples. 4.8 Scalar and interscalar transposition. 4.9 Interscalar transposition and voice leading. 4.10 Combining interscalar and chromatic transpositions. CHAPTER 5. Macroharmony and Centricity 5.1 Macroharmony. 5.2 Small-gap macroharmony. 5.3 Pitch-class circulation. 5.4 Modulating the rate of pitch-class circulation. 5.5 Macroharmonic consistency. 5.6 Centricity. 5.7 Where does centricity come from? 5.8 Beyond "tonal" and "atonal." PART II. History and Analysis CHAPTER 6. The Extended Common Practice 6.1 Disclaimers. 6.2 Two-voice medieval counterpoint. 6.3 Triads and the Renaissance. 6.4 Functional harmony. 6.5 Schumann's Chopin. 6.6 Chromaticism. 6.7 Twentieth-century scalar music. 6.8 The extended common practice. CHAPTER 7. Functional Harmony 7.1 The thirds-based grammar of elementary tonal harmony. 7.2 Voice leading in functional harmony. 7.3 Sequences. 7.4 Modulation and key distance. 7.5 The two lattices. 7.6 A challenge from Schenker. CHAPTER 8. Chromaticism 8.1 Decorative chromaticism. 8.2 Generalized augmented sixths. 8.3 Brahms and Schoenberg. 8.4 Schubert and the major-third system. 8.5 Chopin's tesseract. 8.6 The Tristan Prelude. 8.7 Alternative approaches. 8.8 Conclusion CHAPTER 9. Scales in Twentieth-Century Music 9.1 Three scalar techniques. 9.2 Chord-first composition. A. Grieg's "Droemmesyn," (Vision), Op. 62 no. 5 (1895). B. Debussy's "Fetes" (1899). C. Michael Nyman's "The Mood That Passes Through You" (1993). 9.3 Scale-first composition. A. Debussy's "Des pas sur la neige" (1910). B. Janacek's "On an Overgrown Path," Series II, no. 1 (1908). C. Shostakovich's Fs minor Prelude and Fugue, Op. 87 (1950). D. Reich's "New York Counterpoint" (1985). E. Reich's "The Desert Music," movement 1 (1984). F. The Who's "Can't Explain" (1965) and Bob Seger's "Turn the Page" (1973). 9.4 The Subset Technique. A. Grieg's "Klokkeklang," (Bell Ringing), Op. 54 no. 6 (1891). B. "Petit Airs," from Stravinsky's Histoire du Soldat (1918). C. Reich's "City Life" (1995). D. Stravinsky's "Dance of the Adolescents" (1913). E. The Miles Davis Group's "Freedom Jazz Dance" (1966). 9.5 Conclusion. CHAPTER 10. Jazz. 10.1 Basic jazz voicings. 10.2 From thirds to fourths. 10.3 Tritone substitution. 10.4 Altered chords and scales. 10.5 Bass and upper-voice tritone substitutions. 10.6 Polytonality, sidestepping, and "playing out." 10.7 Bill Evans's "Oleo." 10.8 Jazz as modernist synthesis. CONCLUSION APPENDIX A. Measuring voice-leading size APPENDIX B. Chord geometry: a technical look. APPENDIX C. Discrete voice leading lattices. APPENDIX D. The interscalar interval matrix. APPENDIX E. Scale, macroharmony, and Lerdahl's "basic space." APPENDIX F. Some study questions, problems, and activities. BIBLIOGRAPHY INDEX

About the Author

Dmitri Tymoczko is a composer and music theorist who teaches at Princeton University. His CD Beat Therapy is available from Bridge records.


"As far as I know, the intersection of those who are distinguished composers and those who have published in Science contains one member: the author of this book. If you are interested in tonality in music, you must read it, because it describes by far the most comprehensive theory of what makes tonal music work." --Philip Johnson-Laird, Stuart Professor of Psychology, Princeton University "A Geometry of Music is an epoch-making publication in music theory and will certainly stimulate other new and innovative work in the field. Tymoczko has produced an outstandingly original synthesis of new music theory that unifies quite a large number of separate subfields and realizes the theorist's dream of finding the rational basis for tonality and tonal-compositional practices in music." --Daniel Harrison, Allen Forte Professor of Music Theory and Chair, Yale University Department of Music "A provocative and ingenious melding of music, geometry, and history that promises to change the way that composers, music theorists, and cognitive scientists view music." -- Gary Marcus, Professor of Psychology, New York University and author of Kluge: The Haphazard Evolution of The Human Mind "Tymoczko's A Geometry of Music is an appealingly written, substantial treatise on tonal harmony. The author introduces his original concepts with clarity and fearlessness. Musicologists, musicians, and listeners with an analytical bent will find plenty of ideas to chew on in this intriguing, rewarding book." --Vijay Iyer, musician "Tymoczko confronts with apparent relish the daunting challenge of selling his ideas to a broad audience of theorists, composers, musicians, and students, and his ability to capture the intricacies of complex material while presenting it clearly and comprehensibly is praiseworthy...If the author's way of doing music theory or promulgating his results is not quite like most of the music theory that we have learned and taught, that is hardly a sufficient reason why we should not give his powerful ideas the attention they deserve." --Music Theory Online "A tour de force, a rich and suggestive summation of an exciting new perspective, -a jumping-off point for further explorations. His geometric diagrams provide new kinds of spatialized representations of the aural facts of tonal experience. They may help composers and musicians to 'see' new possibilities within that intricate labyrinth, as well as to bring the old ones to life anew." --Times Literary Supplement "Formidable...The strongest aspect of Tymoczko's book is the case that he gives for voice-leading in the common practice." --Reason Papers

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